3 Answers
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No, Pluto is a so called resonant trans-neptunian object; the orbital period of Pluto is almost exactly 3:2 (1.5) times that of Neptune. This means that every time Pluto nears perihelion and is therefore closest to the Sun and also closest to the orbit of Neptune, Neptune is always at a specific angle (50° according to Wikipedia) in front or behind Pluto.
(See for instance Figure 3 and surrounding text in Jewitt, D., Morbidelli, A., & Rauer, H. (2007). Trans-Neptunian Objects and Comets. Springer.)

Furthermore, the orbits do not actually overlap. Remember that the Solar System is 3-dimensional. Pluto's orbit is substantially tilted relative to Neptune's, and even if they weren't resonant they wouldn't collide because their orbits don't intersect at any point. (Looking at a map of the Solar System from "above", at the two points where the orbits seem to intersect they're at different "heights".)
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Keith ThompsonJan 24 '14 at 21:58

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Quite true, but the fact that it is resonant prevents even perturbations from Neptune, which could cause it to collide with Neptune or (more likely) being ejected from the solar system. That is the reason why there are so many more bodies at resonant orbits than at non-resonant orbits.
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DieudonnéJan 25 '14 at 8:39

No. From 1979 to 1999, Pluto was the eighth planet from the sun. In 1999, it slipped beyond Neptune to become the ninth. But Pluto's 248-year orbit around the sun takes it 17 degrees above and below the plane in which Neptune and the other planets travel.

So their paths don't actually cross as they swap positions. Imagine you are the sun in the middle of your back yard. The fence is Neptune's orbit. You toss a boomerang way out over the neighbor's houses and it comes back, being on both sides of your fence during its travels without hitting the fence. Of course, activity like that can be frowned upon, and in Pluto's case helped lead to its demotion.